Precalc
posted by pat on .
Find the center and vertices of the ellipse.
4x^2 + 16y^2  64x  32y + 208 = 0

4X^2 + 16Y^2  64X  32Y + 208 = 0.
Rearrange variables:
4X^2  64X + 16Y^2  32Y = 208,
Divide both sides by 4:
X^2  16X + 4Y^2  8Y = 52,
Complete the square:
X^2  16X + (16/2)^2 + 4(Y^2  2Y +
(2/2)^2) = 52 + 64 + 4 = 16,
Simplify:
X^2  16X + 64 + 4(Y^2  2Y + 1) = 16.
Write the perfect squares as binomials:
(X  8)^2 + 4(Y  1)^2 = 16,
Divide both sides by 16 and get:
(X  8)^2 / 16 + (Y  1)^2 / 4 = 1.
C(h , k) = C(8 , 1).
a^2 = 16,
a = + 4.
b^2 = 4,
b = +2.
Major Axis:
V1(X1 , Y1), C(8 , 1), V2(X2 , Y2).
h  X1 = a,
X1 = h  a,
X1 = 8  4 = 4.
Y1 = Y2 = K = 1.
X2  h = a,
X2 = h + a,
X2 = 8 + 4 = 12.
Minor Axis:
V4(X4 , Y4)
C(8 , 1)
V3(X3 , Y3)
X3 = X4 = h = 8.
k  Y3 = b,
Y3 = k  b,
Y3 = 1  4 = 3.
Y4  k = b,
Y4 = k + b,
Y4 = 1 + 4 = 5.
Center and Vertices:
C(8 , 1)
V1(4 , 1)
V2(12 , 1)
V3(8 , 3)
V4(8 , 5).
Complete the square: 
CORRECTION:
b = 2,
Y3 = k  b,
Y3 = 1  2 = 1.
Y4 = k + b,
Y4 = 1 + 2 = 3.
V3(8 , 1)
V4(8 , 3).